### expected size of the largest piece

expected size of the largest piece of the loop

what to see.

what to not see.

t math. t probability.

loop

expected size of the largest piece of the loop

what to see.

what to not see.

t math. t probability.

expected size of the largest piece of the loop

what to see.

what to not see.

t math. t probability.

math study

t study. t math. t my.

undergraduate abstract algebra:

over.

abstract linear algebra:

need study.

undergraduate real analysis:

over. (over because i find i can solve exercises of lebesgue measure theory, which is the next step of college real analysis)

complex analysis:

need study.

point set topology:

<-- I think there's no point (unless for practising rigorousness in mathematics) in studying point set theory on its own. (just like there's no point in studying category theory in its own.) so? study only the needed parts when needed. point set topology and category theory is too general and too abstract.

detail's not important.

t study. t math. t my.

undergraduate abstract algebra:

over.

abstract linear algebra:

need study.

undergraduate real analysis:

over. (over because i find i can solve exercises of lebesgue measure theory, which is the next step of college real analysis)

complex analysis:

need study.

point set topology:

<-- I think there's no point (unless for practising rigorousness in mathematics) in studying point set theory on its own. (just like there's no point in studying category theory in its own.) so? study only the needed parts when needed. point set topology and category theory is too general and too abstract.

detail's not important.

bijections from NxN onto N

tag: t math. t function. t bijection. t polynomial. t zakon.

bijections from N x N onto N

f(n,m) = 2^n (2m-1)

g(x,y) = g(s,y) = 1/2 (x+y-1)(x+y) + (1-y)

For f, observe that the expression 2m-1 usually denotes odd numbers. f comes from and gives rise to a multiplication-based decomposition of N into N pieces.

For g, consider s = x+y as fixed and y varing. g comes from and gives rise to a decomposition of N into N intervals.

In fact, g is the famous zig-zag proof of equinumeracy of N x N and N.

Note that g is a polynomial.

Note that, to prove that f and g are bijections, we find meanings in the expressions of f and g. Find meanings and motivations in expressions, formula, formulation, proof, theorem, conjecture, equalities, inequalities

ref:

f from sci.math

g from http://www.trillia.com/distr3/zakon-basic-a4-one.pdf

tag: t math. t function. t bijection. t polynomial. t zakon.

bijections from N x N onto N

f(n,m) = 2^n (2m-1)

g(x,y) = g(s,y) = 1/2 (x+y-1)(x+y) + (1-y)

For f, observe that the expression 2m-1 usually denotes odd numbers. f comes from and gives rise to a multiplication-based decomposition of N into N pieces.

For g, consider s = x+y as fixed and y varing. g comes from and gives rise to a decomposition of N into N intervals.

In fact, g is the famous zig-zag proof of equinumeracy of N x N and N.

Note that g is a polynomial.

Note that, to prove that f and g are bijections, we find meanings in the expressions of f and g. Find meanings and motivations in expressions, formula, formulation, proof, theorem, conjecture, equalities, inequalities

ref:

f from sci.math

g from http://www.trillia.com/distr3/zakon-basic-a4-one.pdf

criteria for being Riemann integrable

tag: math. proof. divide. sci.math. integrable. criteria. using compact. using refinement. using covering. oscillation. envelope. using closure. using point set topology. proving measure zero. history.

f : [0,1] or [a,b] --> [-M,M] or [-1,1]

D = the set of points where f is discontinuous

D_e = the set of points where f is discontinuous by size larger than e somehow.

"f (bounded) is Riemann integrable on [0,1] iff m(D) = 0"

It says f is Riemann integrable iff the set of badly-behaving points D is small.

See Robert Israel's nice summary of a proof (the 'if' part) in which he directly successfully uses D rather than using D_e in constructing the required partition of [0,1]. Rather than to use directly D and K = [0,1]\D, he covers (and replace) D with a slightly bigger open set U and K with V. (compactness of K is used to construct V and finiteness of F1 is used to construct F2). Uses F1 and F2 to find a refinement partition pi. pi is not directly constructed from D and K, but by first covering and replacing D and K both by F1 and F2 (finite collection of open intervals).

learn: flexibility. using refinments. using covering.

concept: covering. refinement. compact.

See Herman Rubin's proof (of both 'if' and 'only if' part at once) in which he uses D_e (in his notation, A_e) rather than using directly D. He uses that 'the closure of A_e is contained in A_(e/2)'. this is for using compactness to reduce to 'finitely many' intervals.

learn: using closure containment.

concept: closure containement. compact.

off topic.

See Robert Israel's 'only if' proof. (a usual and straightforward proof). The proof can be considered 'divide and conquer' twice. He reduces

m(D)=0

to

m(D_e) < (arbitrary small).

First 'divide and conquere', in that D --> D_e.

Second 'divide and conquere', in that 0 --> arbitrary small.

Observe that this is usual in proving something is measure zero.

learn: "take epsilon -> 0". usual way of proving m(something)=0.

See R3769's where the concept of lower envelope and upper envelope is used.

See Dave L. Renfro's post, for the history. an excerpt:

"Riemann's nontrivial contributions on this topic were (a) giving a

necessary and sufficient condition for integrability based on the

behavior of a function, (b) using this condition to prove the

integrability of a function having a dense set of discontinuities,

and (c) putting the focus on the collection of functions that are

integrable according to some notion of integrability, rather than

defining a notion of integrability only in order to rigorously

prove certain desired integrability properties."

tag: math. proof. divide. sci.math. integrable. criteria. using compact. using refinement. using covering. oscillation. envelope. using closure. using point set topology. proving measure zero. history.

f : [0,1] or [a,b] --> [-M,M] or [-1,1]

D = the set of points where f is discontinuous

D_e = the set of points where f is discontinuous by size larger than e somehow.

"f (bounded) is Riemann integrable on [0,1] iff m(D) = 0"

It says f is Riemann integrable iff the set of badly-behaving points D is small.

See Robert Israel's nice summary of a proof (the 'if' part) in which he directly successfully uses D rather than using D_e in constructing the required partition of [0,1]. Rather than to use directly D and K = [0,1]\D, he covers (and replace) D with a slightly bigger open set U and K with V. (compactness of K is used to construct V and finiteness of F1 is used to construct F2). Uses F1 and F2 to find a refinement partition pi. pi is not directly constructed from D and K, but by first covering and replacing D and K both by F1 and F2 (finite collection of open intervals).

learn: flexibility. using refinments. using covering.

concept: covering. refinement. compact.

See Herman Rubin's proof (of both 'if' and 'only if' part at once) in which he uses D_e (in his notation, A_e) rather than using directly D. He uses that 'the closure of A_e is contained in A_(e/2)'. this is for using compactness to reduce to 'finitely many' intervals.

learn: using closure containment.

concept: closure containement. compact.

off topic.

See Robert Israel's 'only if' proof. (a usual and straightforward proof). The proof can be considered 'divide and conquer' twice. He reduces

m(D)=0

to

m(D_e) < (arbitrary small).

First 'divide and conquere', in that D --> D_e.

Second 'divide and conquere', in that 0 --> arbitrary small.

Observe that this is usual in proving something is measure zero.

learn: "take epsilon -> 0". usual way of proving m(something)=0.

See R3769's where the concept of lower envelope and upper envelope is used.

See Dave L. Renfro's post, for the history. an excerpt:

"Riemann's nontrivial contributions on this topic were (a) giving a

necessary and sufficient condition for integrability based on the

behavior of a function, (b) using this condition to prove the

integrability of a function having a dense set of discontinuities,

and (c) putting the focus on the collection of functions that are

integrable according to some notion of integrability, rather than

defining a notion of integrability only in order to rigorously

prove certain desired integrability properties."

"Are a matrix and it's transpose similar?"

three proofs in the sci.math (google groups) thread.

proof 1 : Divide and Conquer Method: rational canonical form and with that minimal polynomial determines companion matrix.

proof 2 : Divide and Conquer Method: Jordan normal form decomposition J = D + N and showing that N is similar to transpose of N by permutation.

Above two are examples of Divide And Conquere that is not Divide Into Cases. (ways: direct sum decomposition. linear sum)

proof 3 : symmetry in computing g_1,...

tag: math. linear. algebra. proof. divide

three proofs in the sci.math (google groups) thread.

proof 1 : Divide and Conquer Method: rational canonical form and with that minimal polynomial determines companion matrix.

proof 2 : Divide and Conquer Method: Jordan normal form decomposition J = D + N and showing that N is similar to transpose of N by permutation.

Above two are examples of Divide And Conquere that is not Divide Into Cases. (ways: direct sum decomposition. linear sum)

proof 3 : symmetry in computing g_1,...

tag: math. linear. algebra. proof. divide

tag: math. definition.

justification of a definition

1. fits mathematical purposes

2. makes the notion precise

justification of a definition

1. fits mathematical purposes

2. makes the notion precise

Proof By Cases: Example from monotonic subsequence theorem

tag: math. proof. case.

"every infinite sequence of real numbers have a monotonic infinite subsequence."

the conclusion has two CASES: having nonincreasing infinite subsequence and having nondecreasing infinite subsequence.

one proof (in Zakon series) starts by dividing two CASES:

case 1: SOME subsequence of the sequence don't have maximum value.

case 2: ALL subsequence of the sequence have max.

SOME or ALL

another proof (in everything2) starts by dividing cases:

case 1: infinitely many maximal index.

case 2: only finitely many maximal index. so that no index is a maximal index from on some N.

, where maximal index is defined as http://everything2.com/index.pl?node_id=983077

Infinite occurence or only finitely many occurence.

another proof starts by dividing:

case 1: unbounded sequence

case 2: bounded sequence.

, then case 2 is more divided into cases,

, as in http://www.puc.edu/Faculty/George_Hilton/id94.htm

unbounded or bounded.

all the three are proofs by cases.

tag: math. proof. case.

"every infinite sequence of real numbers have a monotonic infinite subsequence."

the conclusion has two CASES: having nonincreasing infinite subsequence and having nondecreasing infinite subsequence.

one proof (in Zakon series) starts by dividing two CASES:

case 1: SOME subsequence of the sequence don't have maximum value.

case 2: ALL subsequence of the sequence have max.

SOME or ALL

another proof (in everything2) starts by dividing cases:

case 1: infinitely many maximal index.

case 2: only finitely many maximal index. so that no index is a maximal index from on some N.

, where maximal index is defined as http://everything2.com/index.pl?node_id=983077

Infinite occurence or only finitely many occurence.

another proof starts by dividing:

case 1: unbounded sequence

case 2: bounded sequence.

, then case 2 is more divided into cases,

, as in http://www.puc.edu/Faculty/George_Hilton/id94.htm

unbounded or bounded.

all the three are proofs by cases.

tag: recovery math

My unlisted three posts on February 4:

http://archiver8.blogspot.com/2006/02/ewd975-dijkstras-pythagorean-theorem.html

http://archiver8.blogspot.com/2006/02/thats-not-only-way-largest-example.html

http://archiver8.blogspot.com/2006/02/inductive-proof-example.html

- - - - - - -

link: EWD975 Dijkstra's Pythagorean Theorem

tag: math. auxiliary. construct. formulate.

good example of constructing an auxiliary figure.

good example of symmetric formulation of a given theorem.

pnt: He takes into account that the formulation be made symmetric.

pnt: In constructing the auxiliary figure, Dijkstra considers that the difference alpha+beta-gamma should be constructed and that it should be done in a way that alpha and beta are symmetric. hence that germ of the figure. Lesson is that the auxiliary figure is not pulled out of a magicians hat, but follows naturally from the theorem to be proved.

- - - - - -

an inductive proof: example

link: Wallace-Bolyai-Gerwien Theorem at cut-the-knot

tag: inductive. math. proof. example. divide. step. intermediate. auxiliary.

my notation

a + b : a and b put together.

a = b : a and b are equidecomposable.

the proof of the theorem combines various simple equidecomposability results in these steps:

1. polygon = triangle + triangle + ...

2. triangle = rectangle = square

3. square + square + ... = big square

4. (from 1&2&3) therefore polygon = square

5. (finally) polygon = square = another polygon.

1&3 are examples of "divide and conquer", in particular, decomposing the mathematical object into simpler pieces.

2 are examples of "step by step" or "intermediate object": where rectangle is the intermediat object.

4to5 is an example of introducing "intermediate object", where square is the intermediate object for the proof.

the following three simple equidecomposability results are used in the proof:

* square = rectangle

* rectangle = triangle

* square + square = square

- - - - - - -

That's not the only way: the largest example

link: Pythagorean Theorem and its many proofs at cut-the-knot

54 proofs of the pythagorean theorem.

lesson: there are always other ways to prove a given theorem.

tag: math. alternative. proof. many.

My unlisted three posts on February 4:

http://archiver8.blogspot.com/2006/02/ewd975-dijkstras-pythagorean-theorem.html

http://archiver8.blogspot.com/2006/02/thats-not-only-way-largest-example.html

http://archiver8.blogspot.com/2006/02/inductive-proof-example.html

- - - - - - -

link: EWD975 Dijkstra's Pythagorean Theorem

tag: math. auxiliary. construct. formulate.

good example of constructing an auxiliary figure.

good example of symmetric formulation of a given theorem.

pnt: He takes into account that the formulation be made symmetric.

pnt: In constructing the auxiliary figure, Dijkstra considers that the difference alpha+beta-gamma should be constructed and that it should be done in a way that alpha and beta are symmetric. hence that germ of the figure. Lesson is that the auxiliary figure is not pulled out of a magicians hat, but follows naturally from the theorem to be proved.

- - - - - -

an inductive proof: example

link: Wallace-Bolyai-Gerwien Theorem at cut-the-knot

tag: inductive. math. proof. example. divide. step. intermediate. auxiliary.

my notation

a + b : a and b put together.

a = b : a and b are equidecomposable.

the proof of the theorem combines various simple equidecomposability results in these steps:

1. polygon = triangle + triangle + ...

2. triangle = rectangle = square

3. square + square + ... = big square

4. (from 1&2&3) therefore polygon = square

5. (finally) polygon = square = another polygon.

1&3 are examples of "divide and conquer", in particular, decomposing the mathematical object into simpler pieces.

2 are examples of "step by step" or "intermediate object": where rectangle is the intermediat object.

4to5 is an example of introducing "intermediate object", where square is the intermediate object for the proof.

the following three simple equidecomposability results are used in the proof:

* square = rectangle

* rectangle = triangle

* square + square = square

- - - - - - -

That's not the only way: the largest example

link: Pythagorean Theorem and its many proofs at cut-the-knot

54 proofs of the pythagorean theorem.

lesson: there are always other ways to prove a given theorem.

tag: math. alternative. proof. many.

link test

@D @Firefox.

same.

blummy's BlogThis! button.

a javascript.

longer post.

longer post.

longer post.

longer post.

longer post.

longer post.

resizing.

link test

(link button press test)

test: wait 5 minutes

going to other page. minimizing the window. but i think that was more than 5 minutes, i did go outside for a while maybe.(?)

two post lost? one is: many proofs of pythagorean theorem (from cut the knot) the largest example of "there is always another proof". where did my post go...

@D @Firefox.

same.

blummy's BlogThis! button.

a javascript.

longer post.

longer post.

longer post.

longer post.

longer post.

longer post.

resizing.

link test

(link button press test)

test: wait 5 minutes

going to other page. minimizing the window. but i think that was more than 5 minutes, i did go outside for a while maybe.(?)

two post lost? one is: many proofs of pythagorean theorem (from cut the knot) the largest example of "there is always another proof". where did my post go...